Multiple choice

If $p, q, r$ in harmonic progression and $p$ & $r$ be different having same sign, then the roots of the equation $px^2\, +\, qx\, +\, r\, =\, 0$ are

  1. real and equal.

  2. real and distinct.

  3. irrational.

  4. imaginary.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If p, q, r are in HP, then q = 2pr/(p+r). The discriminant D = q^2 - 4pr = (2pr/(p+r))^2 - 4pr = 4p^2r^2/(p+r)^2 - 4pr = 4pr [pr/(p+r)^2 - 1] = 4pr [ (pr - (p^2 + 2pr + r^2)) / (p+r)^2 ] = -4pr(p^2 + pr + r^2) / (p+r)^2. Since p and r have the same sign, pr > 0, so D < 0, meaning roots are imaginary.

AI explanation

Since p, q, and r are in harmonic progression, their reciprocals 1/p, 1/q, and 1/r are in arithmetic progression, meaning 1/p + 1/r = 2/q. Multiplying by pqr yields qr + pq = 2pr, which rearranges to q = 2pr / (p + r). The discriminant of the quadratic equation is q^2 - 4pr, which substitutes to (4p^2r^2 / (p + r)^2) - 4pr, equaling -4pr(p - r)^2 / (p + r)^2. Since p and r have the same sign but are different, pr is positive and (p - r)^2 is positive, making the discriminant strictly negative and the roots imaginary.

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